Michael: both making the same set of mistakes in their efforts to understand Bayesian reasoning, anthropics, decision theory, etc?
Would you like to elaborate on what those mistakes are (thereby helping out this simple but misguided algorithm)? If there are links to existing sources, that would be enough.
Speaking of measure theory, what probability should we assign to a uniformly distributed random real number on the interval [0, 1] being rational? Something bigger than 0? Maybe in practice we would never hold a uniform distribution over [0, 1] but would assign greater probability to “special” numbers (like, say, 1⁄2). But regardless of our probability distribution, there will exist subsets of [0, 1] to which we must assign probability 0.
The only way I can see around this is to refuse to talk about infinite (or at least uncountable) sets. Are there others?
There are uncountably many possible worlds. Using standard real-number-valued probabilities, we have to assign probability zero to (I think) almost all of them. In other words, for almost all of the possible worlds, the probability of the complement of that possible world is 1.
(Are there ways around this, perhaps using non-real-valued probabilities?)
How much time did you spend trying to come up with predictions from these hypotheses before declaring them unfalsifiable?
Not much; it’s possible that these hypotheses are falsifiable (in the sense of having a likelihood ratio < 1 compared against the other corresponding hypothesis). I was assuming this wasn’t true given only the evidence currently available, but I’d be glad to hear if you think otherwise.
Then why do they say “Science is based on faith too!” in that angry-triumphal tone, rather than as a compliment?
When used appropriately, the “science is based on faith too” point is meant to cast doubt upon specific non-falsifiable conclusions that scientists take for granted: for instance, that the only things that exist are matter (rather than, say, an additional immaterial spirit) or that evolution happens by itself (rather than, say, being directed by an intelligent designer). Scientific evidence doesn’t distinguish between these hypotheses; it’s taken on faith that the first of these is “simpler” and deserves higher prior probability. Maybe these priors are derived from Kolmogorov complexity or something similar, but it still must be taken on faith that those measures are meaningful. (This is, of course, what you recognized when you said, “Every worldview imposes some of its structure on its observations [...].”)
Induction is not logically justified, but you can make a different argument. You could point out that creatures who ignore the apparent patterns in nature tend to die pretty quick. Induction is a behavior that seems to help us stay alive.
Isn’t this argument premised on induction, i.e., things that helped organisms stay alive in the past will help them stay alive in the future?
I would ask the same question as Nominull and Tiiba: Why is a fundamentally mental thing different from a fundamentally physical thing like quarks? If we discovered a spirit in a tree that wasn’t composed of quarks and leptons, is there a reason we couldn’t take that spirit to be a new fundamental particle that behaves in such-and-such a way, just as a down quark is a fundamental particle that behaves in such-and-such a way?
Eliezer: If, on the other hand, a supernatural hypothesis turns out to be true, then presumably you will also discover that it is not inconceivable.
Right. So apart from Occam’s razor, what’s the reason for excluding things that aren’t quarks and leptons from your set of fundamental particles?